# Data-dependent compression of random features for large-scale kernel approximation

arXiv: https://arxiv.org/abs/1810.04249 · HTML: https://mlsanity.com/paper/1810.04249/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2018.

## Built on by (most cited first, 3 of 3)

- [Random matrices in service of ML footprint: ternary random features with no performance loss](https://mlsanity.com/paper/2110.01899/) (2021)
- [Sigma-Delta and Distributed Noise-Shaping Quantization Methods for Random Fourier Features](https://mlsanity.com/paper/2106.02614/) (2021)
- [Decentralized Kernel Ridge Regression Based on Data-Dependent Random Feature](https://mlsanity.com/paper/2405.07791/) (2024)

## Abstract (arXiv)

Kernel methods offer the flexibility to learn complex relationships in modern, large data sets while enjoying strong theoretical guarantees on quality. Unfortunately, these methods typically require cubic running time in the data set size, a prohibitive cost in the large-data setting. Random feature maps (RFMs) and the Nystrom method both consider low-rank approximations to the kernel matrix as a potential solution. But, in order to achieve desirable theoretical guarantees, the former may require a prohibitively large number of features J+, and the latter may be prohibitively expensive for high-dimensional problems. We propose to combine the simplicity and generality of RFMs with a data-dependent feature selection scheme to achieve desirable theoretical approximation properties of Nystrom with just O(log J+) features. Our key insight is to begin with a large set of random features, then reduce them to a small number of weighted features in a data-dependent, computationally efficient way, while preserving the statistical guarantees of using the original large set of features. We demonstrate the efficacy of our method with theory and experiments--including on a data set with over 50 million observations. In particular, we show that our method achieves small kernel matrix approximation error and better test set accuracy with provably fewer random features than state-of-the-art methods.